Wee welcome all those who are interested to join us on Tuesdays, 3:00–4:00 PM, Ryerson Laboratory, Room 358, unless noted otherwise.

Everyone is also invited to attend the Learning Seminar on Geometric Analysis, organized by Prof. Andre Neves. More information on http://www.math.uchicago.edu/~geometric_analysis/learning

Gang Liu, Northwestern University

**
On some recent progress of Yau's uniformization conjecture
:**
Yau's uniformization conjecture states that a complete noncompact Kahler manifold with positive bisectional
curvature is biholomorphic to the complex Euclidean space. We shall discuss some recent progress via the Gromov-Hausdorff
convergence technique.

John Ma, University of British Columbia

**
**** Compactness, finiteness properties of Lagrangian self-shrinkers in R^4 and Piecewise mean curvature flow.
:**
In this talk, we discuss a compactness result on the space of compact Lagrangian self-shrinkers in R^4. When the area is bounded above
uniformly, we prove that the entropy for the Lagrangian self-shrinking tori can only take finitely many values; this is done by deriving a
Lojasiewicz-Simon type gradient inequality for the branched conformal self-shrinking tori. Using the finiteness of entropy values, we construct a
piecewise Lagrangian mean curvature flow for Lagrangian immersed tori in R^4, along which the Lagrangian condition is preserved, area is
decreasing, and the type I singularities that are compact with a fixed area upper bound can be perturbed away in finite steps. This is a Lagrangian
version of the construction for embedded surfaces in R^3 by Colding and Minicozzi. This is a joint work with Jingyi Chen.

Laura Schaposnik, University of Illinois at Chicago

**
On the geometry of branes in the moduli space of Higgs bundles.
:**
After giving a gentle introduction to Higgs bundles, their moduli space and the associated Hitchin fibration,
I shall describe how actions both on groups and on surfaces (anti-holomorphic or of finite groups) lead to families of interesting
subspaces of the moduli space of Higgs bundles (the so-called branes). Finally, we shall look at correspondences between these branes
that arise from Langlands duality, as well as from other relations between Lie groups.

Pei-Ken Hung, Columbia University

**
**** A Minkowski inequality for hypersurfaces in the Anti-deSitter-Schwarzschild manifold
:**
We prove a sharp inequality for hypersurfaces in the
Anti-deSitter-Schwarzschild manifold. This inequality generalizes the
classical Minkowski inequality for surfaces in the three dimensional
Euclidean space, and has a natural interpretation in terms of the Penrose
inequality for collapsing null shells of dust. The proof relies on a
monotonicity formula for inverse mean curvature flow, and uses a geometric
inequality established by Brendle.

Lan-Hsuan Huang, University of Connecticut

**
Geometry of Static Asymptotically Flat 3-Manifolds
:**
A static potential on a 3-manifold reflects the symmetry of the spacetime development and is
also closely related to the scalar curvature deformation.
We discuss how minimal surfaces of 3-manifold interact with the static potential and, as an application,
give a new proof to the rigidity of the Riemannian positive mass theorem.

Yu Li, University of Wisconsin-Madison

**
Ricci flow on asymptotically Euclidean manifolds
:**
In this talk, we prove that if an asymptotically Euclidean (AE) manifold with nonnegative scalar curvature has long
time existence of Ricci flow, it converges to the Euclidean space in the strong sense. By convergence, the mass will drop to
zero as time tends to infinity. Moreover, in three dimensional case, we use Ricci flow with surgery to give an independent
proof of positive mass theorem.

Yannick Sire, Johns Hopkins University

**
Bounds on eigenvalues on Riemannian surfaces
:**
In a joint program with N. Nadirashvili, we developed some tools to prove
existence of extremal metrics in conformal classes for eigenvalues of the Laplace-Beltrami operator on surfaces.
I will describe these results and move to an application to the isoperimetric inequality of the third eigenvalue on the 2-sphere.

Henri Roesch, Duke University

**
Proof of a Null Penrose conjecture using a new Quasi-local Mass
:**
We define an explicit quasi-local mass functional which is non-decreasing along all
foliations (satisfying a convexity assumption) of null cones. We use this new functional to prove the null Penrose conjecture
under fairly generic conditions.

Luca Spolaor, MIT

**
A direct approach to an epiperimetric inequality for Free-Boundary problems
:**
Using a direct approach, we prove a 2-dimensional epiperimetric inequality for the one-phase problem in the scalar and vectorial cases and
for the double-phase problem. From this we deduce the $C^{1,\alpha}$ regularity of the free-boundary in the scalar one-phase and double-phase
problems, and of the reduced free-boundary in the vectorial case, without any restriction on the sign of the component functions. In this talk I
will try to explain the proof of the epiperimetric inequality in the scalar one-phase problem. This is joint work with Bozhidar Velichkov.

Victoria Sadovskaya, Department of Mathematics - Penn State

**Boundedness, compactness, and invariant norms for operator-valued cocycles.:**
We consider group-valued cocycles over dynamical systems with
hyperbolic behavior, such as hyperbolic diffeomorphisms or subshifts
of finite type. The cocycle A takes values in the group of invertible
bounded linear operators on a Banach space and is Holder continuous.
We consider the periodic data of A, i.e. the set of its return values
along the periodic orbits in the base. We show that if the periodic data
of A is bounded or contained in a compact set, then so is the cocycle.
Moreover, in the latter case the cocycle is isometric with respect to
a Holder continuous family of norms. This is joint work with B. Kalinin.

Baris Coskunuser , Boston College

**
Embeddedness of the solutions to the H-Plateau Problem
:**
In this talk, we will give a generalization of Meeks and Yau's embeddedness result on the solutions of the Plateau
problem to the constant mean curvature disks. arXiv:1504.00661

Lu Wang, Department of Mathematics - University of Wisconsin-Madison

**Asymptotic structure of self-shrinkers:**
We show that each end of a noncompact self-shrinker in Euclidean
3-space of finite topology is smoothly asymptotic to a regular cone or a
self-shrinking round cylinder.

Spencer Becker-Kahn, Department of Mathematics - MIT

**Zero Sets of Smooth Functions and p-Sweepouts:**
It is well known that any closed set can occur as the zero set of a smooth function. But in many contexts in analysis, one does not work with arbitrary smooth functions; one works with smooth functions that vanish to only finite order. And the zero set of any such function must have some regularity and structure.
With recourse to an analogy made by Gromov and some recent conjectures of Marques and Neves, I will explain the application of some general results about such zero sets to p-sweepouts (which are p-dimensional families of generalized hypersurfaces that sweepout a smooth manifold in a certain sense). In the course of doing so, I will explain a new result about the regularity of zero sets of smooth functions near a point of finite vanishing order (joint work with Tom Beck and Boris Hanin)..

Jonathan Zhu, Department of Mathematics - Harvard University

**
Entropy and self-shrinkers of the mean curvature flow:**
The Colding-Minicozzi entropy is an important tool for understanding the mean curvature flow (MCF), and is a measure of the complexity of a submanifold. Together with Ilmanen and White, they conjectured that the round sphere minimises entropy amongst all closed hypersurfaces. We will review the basics of MCF and their theory of generic MCF, then describe the resolution of the above conjecture, due to J. Bernstein and L. Wang for dimensions up to six and recently claimed by the speaker for all remaining dimensions. A key ingredient in the latter is the classification of entropy-stable self-shrinkers that may have a small singular set.

Greg Chambers, Department of Mathematics - The University of Chicago

**
Existence of minimal hypersurfaces in non-compact manifolds:**
I prove that every complete non-compact manifold of finite volume admits a minimal hypersurface of finite area. This work
is in collaboration with Yevgeny Liokumovich.

Peter McGrath, Brown University

**
New doubling constructions for minimal surfaces:**
I will discuss recent work (with Nikolaos Kapouleas) on constructions of new embedded minimal surfaces using singular perturbation
'gluing' methods. I will discuss at length doublings of the equatorial S^2 in S^3. In contrast to earlier work of Kapouleas, the catenoidal bridges
may be placed on arbitrarily many parallel circles on the base S^2. This necessitates a more detailed understanding of the linearized problem and
more exacting estimates on associated linearized doubling solutions.

Franco Vargas Pallete, UC Berkeley

**
Renormalized volume:**
The renormalized volume V_R is a finite quantity associated to certain hyperbolic 3-manifolds of infinite volume. In this talk I'll discuss its definition and some properties for acylindrical manifolds, namely local convexity and convergence under geometric limits. If time suffices I'll also discuss some applications and further examples.

Christos Mantoulidis, Stanford University

**
Fill-ins, extensions, scalar curvature, and quasilocal mass:**
There is a special relationship between the Jacobi and the ambient scalar curvature operator. I'll talk about an extremal bending technique that exploits this relationship. It lets us compute the Bartnik mass of apparent horizons and disprove a form of the Hoop conjecture due to G. Gibbons. Then, I'll talk about a derived "cut-and-fill" technique that simplifies 3-manifolds of nonnegative scalar curvature. It is used in studying a priori L^1 estimates for boundary mean curvature for compact initial data sets, and in generalizing Brown-York mass. Parts of this talk reflect work done jointly with R. Schoen/P. Miao.

Hung Tran, UC Irvine

**
Index Characterization for Free Boundary Minimal Surfaces:**
A FBMS in the unit Euclidean ball is a critical point of the area functional among all surfaces with boundaries in the unit sphere, the boundary of the ball. The Morse index gives the number of distinct admissible deformations which decrease the area to second order. In this talk, we explain how to compute the index from data of two simpler problems. The first one is the corresponding problem with fixed boundary condition; and the second is associated with the Dirichlet-to-Neumann map for Jacobi fields. We also discuss applications to a conjecture about FBMS with index 4.

For further information on this seminar or this webpage, please contact Marco Guaraco at
`guaraco(at)math.uchicago.edu`, Lucas Ambrozio at
`lambrozio(at)math.uchicago.edu` or
Rafael Montezuma at `montezuma(at)math.uchicago.edu`